Calculate a Recursion

Calculator for recursions with two up to five start values. For one start value, see iteration. Here, a recursion is a repeated calculation with several previously computed values. As recursion variables in the formula, v for r(n-1), w for r(n-2), x for r(n-3), y for r(n-4) and z for r(n-5) are used. Only these variables v, w, x, y and z may occur in the recursion term, if the according number of start values is set. The basic arithmetic operations + - * / are allowed, as well as the power function pow(), like pow(2#z) for 2^z. Further allowed functions are sin(), cos(), tan(), asin(), acos(), atan(), log() for the natural logarithm and fac() for the factorial. Also, the constants e and pi can be used.

Example: r = v + w with two start values r(0)=1 and r(1)=1 is the Fibonacci sequence. There, a new value is calculated from the sum of the two previous values. The Fibonacci sequence is probably the most famous mathematical definition of recursion.

The term recursion comes from the Latin recurrere, meaning to run back. It is used for something that goes back to itself, i.e. goes back to previous states and takes something from there. The mathematical meaning of the recursive definition used here is closely related to iteration, but more comprehensive than iteration in that several start and repetition conditions are taken into account. There is a joke that says that in order to understand recursion, you first have to understand recursion. This is already a pretty good description of recursion in general.
Recursion plays an important role in programming, in the form of functions that call themselves. These are often associated with short source code and are considered elegant, but are also often difficult to understand and therefore pose an increased risk of endless loops.